On non-existence of Nash equilibrium of M person game with pure strategy for delivery services

A competitive market model is considered with M suppliers and N customers, where each supplier provides a homogeneous service such as delivering propane gas cylinders and has to offer a uniform price upon delivery to all customers. Given a price upper bound U, the model is formulated as an M person game with pure strategy. It is shown that the M person game has the unique Nash equilibrium if and only if each customer can be serviced by at most one supplier. Furthermore, this unique Nash equilibrium is peculiar in that all suppliers adopt the same upper bound price U. In general, the M person game does not have any Nash equilibrium. For such a case, it is demonstrated that the suppliers continue to exercise their price strategies in a cyclic manner indefinitely.